Complexity
Volume 2019 (2019), Article ID 9053858, 11 pages
https://doi.org/10.1155/2019/9053858

Research Article

Homeomorphism Mapping Based Neural Networks for Finite Time Constraint Control of a Class of Nonaffine Pure-Feedback Nonlinear Systems

Jianhua Zhang,1,2 Quanmin Zhu,³ Yang Li,1,2 and Xueli Wu1,2

¹Hebei University of Science and Technology, Shijiazhuang, Hebei 050018, China
²Hebei Provincial Research Center for Technologies in Process Engineering Automation, Shijiazhuang, Hebei 050018, China
³Department of Engineering Design and Mathematics, University of the West of England, Coldharbour Lane, Bristol BS16 1QY, UK

Correspondence should be addressed to Jianhua Zhang; jianhuazhang@aliyun.com

Received 15 February 2019; Revised 2 April 2019; Accepted 24 April 2019; Published 13 May 2019

Guest Editor: Chun Wei

Copyright © 2019 Jianhua Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper proposes a new scheme for solving finite time neural networks adaptive tracking control issue for the nonaffine pure-feedback nonlinear system. The procedure, based on homeomorphism mapping and backstepping, effectively deals with constraint control and design difficulty induced by pure-feedback structure. The most outstanding novelty is that finite time adaptive law is proposed for training weights of neural networks. Furthermore, by combining finite time adaptive law and Lyapunov-based arguments, a valid finite time adaptive neural networks controller design algorithm is presented to ensure that system is practical finite stable (PFS) rather than uniformly ultimately bounded (UUB). Because of using the finite time adaptive law to training weights of neural networks, the closed-loop error system signals are in assurance of bounded in finite time. Benchmark simulations have well demonstrated effectiveness and efficiency of the proposed approach.

1. Introduction

During the past decades, lots of studies have been done to solve neural networks (NNs) control and improve the adaptive law for training neural networks weights [1–3]. Among these works, the core is NNs which are used as online approximation functions for the unknown nonlinearities, due to their inherent approximation capabilities [4, 5]. Almost all the neural adaptive control designs and stability analyses are Lyapunov uniformly ultimately bounded (UUB) results [5]; based on the Krasovskii–LaSalle invariance principle, it is challenging to establish a generalized powerful framework for neural control [6–8], even though it has been used to get sufficient conditions for smooth stabilization for closed-loop systems [9–11].

Traditionally, Quadratic Lyapunov Functions are in common used to construct for analysis and control design of linear and nonlinear even time delay dynamic systems [12, 13]. There also exist some other formats of Lyapunov functions, such as integral Lyapunov function [14], barrier Lyapunov function [15, 16], and vector Lyapunov functions [17, 18]. Such attempts have enhanced Lyapunov function applications in control system design [19], for example, integral Lyapunov function for controller singularity problems and barrier Lyapunov function for output constraint problems. Nonlinear systems differ from linear system, because the complexity of the relation between states and input and therefore numerical techniques (open solutions) have played a significant role in controller design process [20–22]. Young’s Inequality as an important lemma has been widespread used to realize UUB (uniformly ultimately bounded) in neural network based adaptive control [23].

In terms of complex of system structure such as pure-feedback and nonaffine, mean value theorem [24] is popular to use to eliminate. Due to its universal approximation capability for nonlinear systems, neural network has been extensively utilized to achieve modelling. Many elegant model structure design techniques, such as strict-feedback, pure-feedback, affine form, and nonaffine systems [3, 25, 26], have been obtained based on backstepping procedure and Lyapunov synthesis.

Based on virtual control and iterative learning control [25], backstepping procedure provided an important methodology to control affine or nonaffine, strict-feedback, or pure-feedback form SISO nonlinear uncertain systems, which are split into several subsystems. In every subsystem, neural networks approximate the unknown nonlinear systems with adaptive law, based on Lyapunov stability theory, then the systems achieve the asymptotic stability or uniformly ultimately bounded stable. In order to better exploit the application of adaptive neural networks [27, 28], finite time stable results are more meaningful for uncertain system [29, 30], high order stochastic nonlinear system [31], and interconnected nonlinear system [32].

It should be noted that the research on finite time neural networks control is still in a very beginning stage. Although there exist a few of adaptive finite control algorithms which have been developed for some special systems and bench tested with simulations and applications [33–35], it is very hard to extend the research results to the general systems or algorithms. The key issue is how to systematically obtain finite time adaptive law of neural networks [9, 36] weight from finite time convergence of close loop systems [37]. In regard to neural network based such adaptive control to the authors’ best knowledge, there are a few results about finite time adaptive neural networks control. That is because it is not easy to design the finite time neural networks adaptive controller and lack of relevant inequality skills to finish finite time stability analysis. There has been any reference to show finite time adaptive algorithm for weights of NNs having been expanded to solve the problems of finite control for pure-feedback nonaffine nonlinear systems.

Motivated by the above critical analyses, we investigated neural backstepping control via finite time adaptive algorithm. At the first stage, the homeomorphism mapping is proposed in place of barrier Lyapunov functions and commonly utilized in traditional constraints control. Then with finite adaptive algorithm, adaptive neural networks backstepping control procedure is designed for nonaffine pure-feedback nonlinear systems. Sufficient condition is given via Lyapunov finite time stability theory to make sure all the signals in the closed-loop system are global uniformly bounded in finite time and the tracking error is remained in a bounded compact set. The main contributions of this paper can be summarized as follows:

(1)This paper is the first study to propose the finite time adaptive law based on Young’s inequality skill for adaptive neural networks control. The main advantages are that adaptive neural networks controls achieve finite time stabilization including weights of neural networks and closed-loop control system uniformly ultimately bounded stable in finite time.(2)To keep full state constraints in a bounded compact set, the homeomorphism mapping is proposed in place of barrier Lyapunov functions and commonly utilized in traditional constraints control.(3)New finite time adaptive neural networks techniques, including virtual control and finite adaptive law, are introduced for ensuring the successful design of the finite time adaptive neural networks backstepping control.

The rest of this paper is organized as follows. Section 2 gives problem formulation and preliminaries, including some definitions and necessary inequality with lemmas. Nonaffine pure-feedback nonlinear systems finite time adaptive neural networks control is considered in Section 3, and a new finite time adaptive law is developed for training neural networks. Lyapunov finite time stable theory is used to guarantee the closed-loop system signals with prescribed performance in finite time. Simulation study is presented in Section 4 to indicate efficiency and effectiveness of the procedure. The conclusion and future works are discussed in Section 5.

2. Problem Description and Preliminaries

In this section, problem description, the notation, basic definitions, and lemmas on the system stability are presented. Consider the following nonlinear system:

where , , , are state variable, control input, output, and initial condition, respectively, and , are nonlinear smooth functions.

The ideal output is ; therefore, the objective is to design an adaptive state feedback to ensure the output tracking and controller in the form

where indicate the estimated values of unknown parameters and the indicate the ideal values. For system (1) with the homeomorphism mapping,

propose to transform the system to realize the constraints control, such as , and , , and then system (1) changes into

where , , , indicate the state, control, output, and initial condition, respectively, and are nonlinear smooth functions. For the homeomorphism mapping, both and exist, and specially, hold.

The adaptive controller in the form

and the output of system (4) track the .

If controller (5) is designed based on system (4), then put the controller into system (1); the states are constraint where , because the states and satisfy the of the homeomorphism mapping relation (3).

Remark 1. There are lots of researches about constraint control [15, 38, 39], which employ the barrier Lyapunov functions to realize constraints. The controller design is relatively complicated because of the complexity of derivation of barrier Lyapunov functions. The homeomorphism mapping skill is easy to achieve, as the mapping is used for transformation of states, which is independent for the controller design. After that, a new model emerged, then based on the new model, all control methods can be used, such as sliding model control [40], adaptive control [26, 41], and especially neural networks control which applied without model information. Such homeomorphism mapping is easy to choose as

which satisfy and , respectively.

A continuous nonaffine pure-feedback nonlinear system can be described as follows:

where , , , indicate the state, control, output, and initial condition, respectively, and , are smooth nonlinear functions.

Assumption 2. The nonlinear function in system (1) is as follows:

when .
Based on homeomorphism mapping (7), nonlinear pure-feedback nonaffine system can be described as follows:

where , , , , and indicate the state, control, output, and initial condition, respectively, and are nonlinear smooth functions.

Assumption 3. Nonlinear function in system (9) is as follows:

when .
Notation. Throughout this paper, represents the matrix, , rational number , and matrix, matrix denote element-by-element powers, , denote the transposition of matrix , such as . indicate ideal weight, estimated weight, and error between ideal and estimated weight.
In brief, the following radial basis function (RBF) NN is used to approximate the continuous function over a compact set

where input , weight vector , and node vector , with the element being chosen as the commonly used Gaussian function as below

where , the center of the receptive field, and is the width of the Gaussian function.
It has been proven that RBF NN can approximate any continuous function over a compact set as

where is the ideal NN weight and is the NN approximation error.

Definition 4. Consider system , where is a state vector and is the input vector. The solution is practical finite stable (PFS) if for all , there exist and , such that , for all .

Lemma 5. Consider system ; suppose that there exist continuous function and scalars , and such that

Then, the trajectory of system is PFS.

Proof. There exists a scalar such that inequality (15) can be expressed as

Clearly, if . According to Lemma 5, the decrease of in finite time drives the trajectories of the closed-loop system into . Therefore, the trajectories of the closed-loop system are bounded in finite time as

where . And the time needed to reach (17) is bounded as

where is the initial value of .

Lemma 6 ([42], Young’s inequality). For any constant , the following inequality holds:

where , , and .

Remark 7. Based on Young’s inequality, then the following inequalities hold:

There are two key inequalities in adaptive neural networks finite time control.

3. Control Design and Stability Analysis

Consider system (9), using the coordinate transform as

where is the virtual control and the ideal virtual control are

In the first step, consider system

Then it has

Based on ideal virtual control law (23), choose the NN to approximate the nonlinear system , , and is compact set

Therefore

Choose the practical virtual control law

Choose the adaptive law

where and is positive constant design parameters.

Then based on (27) and (28)

let

therefore

where

Choose Lyapunov candidate function

then

Based on inequalities from Lemma 6 and Remark 7, the following hold:

and

Then based on (35), it gives

then it has

where

The th step

Choose the virtual control law (23), choose the NN to approximate the nonlinear system , , and is compact set

where

Based on the system,

Choose the practical virtual control law

Choose the adaptive law

where and is positive constant design parameters. Then

where

Choose Lyapunov candidate function

then

Based on Lemma 6 and Remark 7, the following inequalities hold:

and

Then based on (49), it yields

And

where

The th step is the most important step. Based on the system

Choose the NN to approximate the nonlinear system , , and is compact set

where

Based on system

Choose practical virtual control law

Choose adaptive law

where and are positive constant design parameters, then

where

From the inequality

choose Lyapunov candidate function

Then

Based on the lemma, the following inequalities hold:

and

Then, it gives

then

where

The th step is the most important step.

Based on system

Choose the NN to approximate the nonlinear system , , and is compact set

where

Based on system

Choose adaptive law

where and are positive constant design parameters.

then

where

Choose Lyapunov candidate function

then

From the inequality

and

we have

Then

where

Choose Lyapunov candidate functions

then based on (39), (54), (70), and (86), it has

where

Theorem 8. Under Assumptions 2 and 3 and Lemmas 5 and 6, consider nonlinear system (9) for which the model dynamics is approximated by neural networks (26), (42), (57), and (74), the control law (78) with the virtual control (28), (45), and (60), and adaptive laws (29), (46), (61), and (77), then the following statements hold:

(1)All the signals of the closed-loop system, including , remain bounded all the time.(2)The closed-loop signal converges to a compact set defined by

 in finite time, where are constants related to the design parameters.(3)The finite time is given by

 where are constants related to the design parameters and is constant related to Lyapunov candidate functional.

Proof. Considering Lyapunov function candidate (88) and (89), then is bounded in finite time. Therefore, it can conclude that, for all , the error signals , are PFS.

4. Simulation Examples

In this section, simulation examples show the validity and feasibility of the proposed NNs adaptive control design approach.

4.1. Mathematical Example

Consider the following dynamic system:

where , , , and is state, control input, control output, and ideal output, respectively. The control objective of the proposed design method is to make the system output follow the desired reference signal . Select the controller and virtual control as follows:

The weight of NNs is updated by

with , , and ; the initialization of the variables is selected by , , and .

Applying finite time NNs control method, the simulation results are shown by Figures 1–5. Figure 1 shows the trajectory of output and desired reference trajectory. Figure 2 shows the trajectory of controller. Figures 3 and 4 show the trajectory of NNs and function which approximated by NNs. Compared with [3], the method in this paper is more generally used, and both output and weight of NNs are convergence in finite time.

Figure 1: Trajectories of .

Figure 2: Trajectory of control input .

Figure 3: Trajectories of , .

Figure 4: Trajectories of , .

Figure 5: Trajectories of error system.

4.2. Plate-Ball Example

Consider a Plate-Ball model [43] in form of

where is ball mass, is ball moment of inertia, is ball radius, are plate inclination angle in the or direction, are contain the centrifugal torque resulting from rotation and other modelling error, and , are ideal trajectory of .

Based on homeomorphism mapping technique, choose

then is new state variable, and therefore all the time. About new state variable, we have

and

Let , then the above equations can be changed as

then control can be designed by regular control method based on Lyapunov theory, state variable is not restricted, and is restricted by reference signal and bound .

Choose as unknown function, reference signal as sinusoidal signal, and NNs control. The simulation result can be obtained by Figures 6 and 7 about with cosine signal as reference signal. Figure 8 indicates trajectories of ball.

Figure 6: Trajectories of and sinusoidal signal.

Figure 7: Trajectories of and cosine signal.

Figure 8: Trajectories of ball.

5. Conclusions

For technique completeness, the neural networks enhanced finite time constraint control framework for a class of uncertain nonaffine pure-feedback nonlinear systems has developed analysis/proof, design, algorithms, and computational experiments. For reference of neurocontrol, this study attempts to strengthen neural network function to deal with complex nonlinear uncertain system control with different design constraints, which confirms again that the universal approximation property of neural networks feasibly effectively integrated with control system specifications. In line with further expansion of the study, some issues have been identified on the research card, such as how to design better homeomorphism mapping to realize the constraint control, how to extend the general order in finite time adaptive neural networks control, and fast computation algorithms for online computation.

Data Availability

No data were used to support this study.

Conflicts of Interest

The funding did not lead to any conflicts of interest regarding the publication of this manuscript. There are not any other possible conflicts of interest in the manuscript.

Acknowledgments

This work is partially supported by the National Nature Science Foundation of China under Grants 61273188 and 61473312 and Taishan Scholar Construction Engineering Special Funding, Shandong, China. This work is supported by Hebei Province Nature Fund Grant F2015208128 and Project in Hebei Province Department of Education QN20140157 and BJ2016020.

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